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We all like hexapods, from the Centaur to the Griffin. A large problem with the realism of the hexapodal body plan is that there are not many reasons for a creature to have them and is still worth the energy they put into using them.
What evolutionary benefits does a vertebral animal gain for having a hexapodal body plan?
Six limbs adds quite a few possible advantages that you can't get with four limbs. It allows each set of limbs to be specialized for particular purposes. Consider the following body plans:
Centaur plan: Four legs for running, two arms for grasping, fighting, and using tools. Sure, we humans can manage with a bipedal stance, but it comes with a number of costs - diminished stability, back problems, and terrible childbirth. A grazing centauroid might be able to use their hands to pick or carry plants while keeping their head upright and on the lookout for predators. A predatory centaur-lion could run at full speed and use their front claws for slashing at prey repeatedly without slowing down.
Griffin plan: Wings plus a catlike body. Great for a predatory lifestyle - sure, our tetrapod birds can fly, but they have to deal with awkwardly using their feet for standing, fighting, and holding things. Of course flying griffins would need to be small, like birds, but even larger ones could control their pounces in mid-air by gliding, making them better hunters.
Pixie plan: Wings plus feet plus grasping hands. This would allow the creature to walk or run without burning the energy it would need for flight, fly when it needs to move quickly or reach high places, and use its hands to either fight or carry food back to its nest.
Jumping plan: Frogs and kangaroos are great at jumping, but pretty bad at walking normally, and hopping everywhere burns a lot of energy. Imagine a vertebrate version of a grasshopper - it could use its front four legs for running normally while keeping its back legs raised, then use its back legs for jumping when it needs the extra speed. A prey animal with this body plan could also use its back legs for delivering powerful kicks to a pursuer's face while using its front four legs for running.
Yes, six legs has some disadvantages - extra energy for growth and maintenance - but that doesn't mean they'd be useless or redundant. Using the same limbs for multiple purposes tends to mean they are less effective at their jobs than if each limb was specialized for one purpose, and more limbs means more capacity for specialization. If our early fish ancestors had six limbs for whatever reason, we'd be just as likely to re-purpose them as we would be to lose them.
Hexapeds are superior to quadrupeds and bipeds. Lets take the centaur as an example. Its 4 base legs provide lots of stability, which exceeds that of any biped. This is why we build tables with 4 legs, not 2. Also, 4 legged animals make up the fastest land animals on Earth.
Humans, which are bipeds, use their upper limbs - arms - to perform fine manipulations. Other primates do this as well. Other advantages of upright hexapods is the ability to see farther. That is one of the reasons humans went from quadrupedal to bipedal.
Although not in the same body shape as centaurs and griffins, insects are hexapods and make up up 80% of Earth's species and the most of the land's biomass. Having 6 limbs is working quite well for them.
Here are a list of the pros:
6 Legs gives you greater stability. The organism would require less brain power to stop itself from falling over while moving normally, and would be able to keep 3 legs on the ground at any given time. Presumably, this would also give them greater traction.
Losing a limb wouldn't make them so handicapped.
And these are the things that cancel out all the benefits and make it unlikely:
Extra weight and the need for increased cortical representation and muscle control on the extra limbs, which would presumably be load bearing, may be a problem
The heart would need to be stronger purely for the purpose of pumping blood through those extra legs, but since the legs taking a toll on the heart could cause this to actually not give any evolutionary benefit
Unless your hexapod body plan was part of the ancestry of your species, it's super unlikely for this to occur. Imagine that for some reason, creature A (which normally has 4 legs) evolved an extra pair of legs. It requires a larger heart, and an adapted brain, as well as some nerve cord changes to control its 3rd pair of legs. How does this help the creature have more reproductive success? Sure, it's less likely to die in a rocky environment (stability), or it can run faster on plains (maybe), but how would this 6 legged creature reproduce with the 4 legged version of this creature? Mutations need many steps - it's unlikely for 2 legs to come out of nowhere, and there are no reasons why an extra pair of legs would make the creature better for reproduction.
They are still able to function if they loose a limb. For example, a primitive cripple might die if their leg were eaten by a lion because they couldn't hunt or forage any more. If a 6 legged (or 4 legged) animal lost a leg, it could still function about 80% capacity.
